[Paper Review] Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity

Pengzi Miao|ArXiv.org|Sep 18, 2003

Black Holes and Theoretical Physics17 references3 citations

TL;DR

This paper investigates the variational effect of boundary mean curvature on the ADM mass in asymptotically flat Riemannian manifolds, showing that for a domain with quasi-convex boundary, decreasing the boundary mean curvature below the intrinsic value allows a reduction in ADM mass. The key contribution is proving that Bartnik's geometric boundary condition—equality of mean curvature across the boundary in minimal mass extensions—holds under these conditions.

ABSTRACT

We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the extension is dominated by but not identically equal to the one determined by the given domain, we can decrease its ADM mass while raising its boundary mean curvature. Thus our analysis implies that, for a domain with quasi-convex boundary, the geometric boundary condition holds in Bartnik's minimal mass extension conjecture \cite{Bartnik_energy}.

Motivation & Objective

• To analyze how variations in boundary mean curvature affect the ADM mass of asymptotically flat manifolds.
• To investigate the geometric boundary condition in Bartnik's minimal mass extension conjecture.
• To establish that for a domain with quasi-convex boundary, the minimal mass extension requires matching mean curvature across the boundary.
• To extend prior results on piecewise smooth metrics and positive mass theorems to include boundary curvature variations.
• To characterize the structure of mass-minimizing extensions, particularly their scalar curvature and static metric properties.

Proposed method

• Uses a variational approach to deform the metric in the exterior region while preserving asymptotic flatness and non-negative scalar curvature.
• Applies a one-parameter family of conformal deformations $ g_t = v_t^{\frac{4}{n-2}} g $ with $ v_t = 1 + t(u - 1) $, where $ u $ solves a Laplace-type equation with boundary condition $ u = 1 $ on $ \Sigma $.
• Employs the strong maximum principle to show that decreasing $ u $'s normal derivative reduces ADM mass.
• Analyzes the behavior of the boundary mean curvature under conformal deformation using the formula $ H(g_t) = H(g) + \frac{2}{n-2} \frac{\partial v_t}{\partial \vec{n}} $.
• Applies the positive mass theorem and scalar curvature deformation techniques to rule out non-zero scalar curvature in mass-minimizing extensions.
• Uses the static metric deformation result of Corvino to show that mass-minimizing extensions must be static and scalar-flat.

Experimental results

Research questions

• RQ1 Can the ADM mass of an asymptotically flat manifold be reduced by increasing the boundary mean curvature while keeping the interior metric fixed?
• RQ2 Under what geometric conditions does Bartnik's minimal mass extension conjecture imply that the boundary mean curvature must match across the interface?
• RQ3 What is the role of scalar curvature and static metric structure in mass-minimizing extensions?
• RQ4 How does the boundary mean curvature influence the existence and regularity of minimal mass extensions?
• RQ5 Can a mass-minimizing extension exist with non-zero scalar curvature, or must it be scalar-flat and static?

Key findings

• For a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature is strictly less than the intrinsic value, the ADM mass can be decreased by increasing the boundary mean curvature.
• The geometric boundary condition $ H(\Sigma, g_-) = H(\Sigma, g_+) $ holds in Bartnik's minimal mass extension conjecture for domains with quasi-convex boundary.
• Any minimal mass extension must be scalar-flat and static in the interior, as non-zero scalar curvature leads to a contradiction via mass reduction under conformal deformation.
• The mass-minimizing sequence can be reparametrized so that the boundary mean curvature monotonically increases to the intrinsic value $ H(\Sigma, g_-) $.
• The Hawking mass of the boundary $ \Sigma $ in the mass-minimizing sequence monotonically decreases to the Hawking mass in the original domain.
• The existence of a minimal mass extension implies it must satisfy the static metric equation and the boundary mean curvature matching condition.